Acceptance set

In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures.

Mathematical Definition

Given a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$, and letting ${\displaystyle L^{p}=L^{p}(\Omega ,{\mathcal {F}},\mathbb {P} )}$ be the Lp space in the scalar case and ${\displaystyle L_{d}^{p}=L_{d}^{p}(\Omega ,{\mathcal {F}},\mathbb {P} )}$ in d-dimensions, then we can define acceptance sets as below.

Scalar Case

An acceptance set is a set $A$ satisfying:

1. ${\displaystyle A\supseteq L_{+}^{p}}$
2. ${\displaystyle A\cap L_{--}^{p}=\emptyset }$ such that ${\displaystyle L_{--}^{p}=\{X\in L^{p}:\forall \omega \in \Omega ,X(\omega )<0\}}$
3. ${\displaystyle A\cap L_{-}^{p}=\{0\}}$
4. Additionally if $A$ is convex then it is a convex acceptance set
   1. And if $A$ is a positively homogeneous cone then it is a coherent acceptance set^[1]

Set-valued Case

An acceptance set (in a space with $d$ assets) is a set ${\displaystyle A\subseteq L_{d}^{p}}$ satisfying:

1. ${\displaystyle u\in K_{M}\Rightarrow u1\in A}$ with $1$ denoting the random variable that is constantly 1 $\mathbb {P}$-a.s.
2. ${\displaystyle u\in -\mathrm {int} K_{M}\Rightarrow u1\not \in A}$
3. $A$ is directionally closed in $M$ with ${\displaystyle A+u1\subseteq A\;\forall u\in K_{M}}$
4. ${\displaystyle A+L_{d}^{p}(K)\subseteq A}$

Additionally, if $A$ is convex (a convex cone) then it is called a convex (coherent) acceptance set. ^[2]

Note that $K_M = K \cap M$ where $K$ is a constant solvency cone and $M$ is the set of portfolios of the $m$ reference assets.

Relation to Risk Measures

An acceptance set is convex (coherent) if and only if the corresponding risk measure is convex (coherent). As defined below it can be shown that $R_{{A_{R}}}(X)=R(X)$ and $A_{{R_{A}}}=A$.

Risk Measure to Acceptance Set

• If $\rho$ is a (scalar) risk measure then $A_{{\rho }}=\{X\in L^{p}:\rho (X)\leq 0\}$ is an acceptance set.
• If $R$ is a set-valued risk measure then $A_{R}=\{X\in L_{d}^{p}:0\in R(X)\}$ is an acceptance set.

Acceptance Set to Risk Measure

• If $A$ is an acceptance set (in 1-d) then $\rho _{A}(X)=\inf\{u\in {\mathbb {R}}:X+u1\in A\}$ defines a (scalar) risk measure.
• If $A$ is an acceptance set then $R_{A}(X)=\{u\in M:X+u1\in A\}$ is a set-valued risk measure.

Examples

Superhedging price

The acceptance set associated with the superhedging price is the negative of the set of values of a self-financing portfolio at the terminal time. That is

$A = \{-V_T: (V_t)_{t=0}^T \text{ is the price of a self-financing portfolio at each time}\}$.

Entropic risk measure

The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is

$A=\{X\in L^{p}({\mathcal {F}}):E[u(X)]\geq 0\}=\{X\in L^{p}({\mathcal {F}}):E\left[e^{{-\theta X}}\right]\leq 1\}$

where $u(X)$ is the exponential utility function.^[3]

References

1. Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk". Mathematical Finance. 9 (3): 203–228. doi:10.1111/1467-9965.00068.
2. Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk" (pdf). SIAM Journal on Financial Mathematics. 1 (1): 66–95. doi:10.1137/080743494. Retrieved August 17, 2012.
3. Follmer, Hans; Schied, Alexander (October 8, 2008). "Convex and Coherent Risk Measures" (pdf). Retrieved July 22, 2010.